It is customary to assume that random effects are normally distributed.
Skrondal and Rabe-Hesketh (2004, Section 14.2) consider a measurement error problem, and
compare the following two models:

Random effects normally distributed

Non-parametric model for the random effects

A description of the model and data is given here: skewed_re.pdf. The non-parametric model 2)
indicates that the random effects distribution is skewed to the right.

In this example we show: 1) how to implement the model with normal random effects in ADMB-RE (diet.tpl) and 2)
how to modify the the program to obtain skewed random effects (diet_sk.tpl).
Only a small number of changes are needed to modify the ADMB-RE code to
implement the skewed random effects.

Results

By looking at the result files (diet.par and diet_sk.par) we observe the following:

The estimated parameters under the normal model match very closely the estimates in
Table 14.1 of Skrondal and Rabe-Hesketh (2004).

The log-likelihood value for the normal model is -1372.35, while the log-likelihood for the model with
skewed random effects is -1326.49. Hence, given that the skewed model only contains one extra parameter,
it gives a much better fit to data.